let x1, x2 be FinSequence of COMPLEX ; :: thesis: ( len x1 = len x2 implies (x1 + x2) *' = (x1 *' ) + (x2 *' ) )
reconsider x9 = x1 as Element of (len x1) -tuples_on COMPLEX by FINSEQ_2:110;
reconsider y9 = x2 as Element of (len x2) -tuples_on COMPLEX by FINSEQ_2:110;
reconsider x3 = x1 *' as Element of (len (x1 *' )) -tuples_on COMPLEX by FINSEQ_2:110;
reconsider x4 = x2 *' as Element of (len (x2 *' )) -tuples_on COMPLEX by FINSEQ_2:110;
assume A1: len x1 = len x2 ; :: thesis: (x1 + x2) *' = (x1 *' ) + (x2 *' )
then A2: len (x1 + x2) = len x1 by Th6;
A3: ( len x1 = len (x1 *' ) & len x2 = len (x2 *' ) ) by Def1;

A4: now

let i be Nat; :: thesis: ( 1 <= i & i <= len ((x1 + x2) *' ) implies ((x1 + x2) *' ) . i = (x3 + x4) . i )
A5: i in NAT by ORDINAL1:def 13;
assume that
A6: 1 <= i and
A7: i <= len ((x1 + x2) *' ) ; :: thesis: ((x1 + x2) *' ) . i = (x3 + x4) . i
A8: i <= len (x1 + x2) by A7, Def1;
hence ((x1 + x2) *' ) . i = ((x1 + x2) . i) *' by A6, Def1
.= ((x9 . i) + (y9 . i)) *' by A1, A5, Th18
.= ((x1 . i) *' ) + ((x2 . i) *' ) by COMPLEX1:118
.= ((x1 *' ) . i) + ((x2 . i) *' ) by A2, A6, A8, Def1
.= ((x1 *' ) . i) + ((x2 *' ) . i) by A1, A2, A6, A8, Def1
.= (x3 + x4) . i by A1, A3, A5, Th18 ;
:: thesis: verum

end;

len ((x1 *' ) + (x2 *' )) = len x1 by A1, A3, Th6;
then len ((x1 + x2) *' ) = len ((x1 *' ) + (x2 *' )) by A2, Def1;
hence (x1 + x2) *' = (x1 *' ) + (x2 *' ) by A4, FINSEQ_1:18; :: thesis: verum